Flat and Multimodal Likelihoods and Model Lack of Fit in Curved Exponential Families

摘要

It is well known that curved exponential families can have multimodal likelihoods. We investigate the relationship between flat or multimodal likelihoods and model lack of tit, the latter measured by the score (Rao) test statistic W_U of the curved model as embedded in the corresponding full model. When data yield a locally flat or convex likelihood (root of multiplicity > 1, terrace point, saddle point, local minimum), we provide a formula for W_U in such points, or a lower bound for it. The formula is related to the statistical curvature of the model, and it depends on the amount of Fisher information. We use three models as examples, including the Behrens-Fisher model, to see how a flat likelihood, etc. by itself can indicate a bad fit of the model. The results are related (dual) to classical results by Efron from 1978.